# Cosine in Terms of Haversine

## Theorem

$\cos \theta = 1 - 2 \hav \theta$

where $\cos$ denotes cosine and $\hav$ denotes haversine.

## Proof

 $\ds \hav \theta$ $=$ $\ds \dfrac 1 2 \paren {1 - \cos \theta}$ Definition of Haversine $\ds \leadsto \ \$ $\ds 2 \hav \theta$ $=$ $\ds 1 - \cos \theta$ $\ds \leadsto \ \$ $\ds \cos \theta$ $=$ $\ds 1 - 2 \hav \theta$

$\blacksquare$